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Properties

Label 1-161-161.82-r1-0-0
Degree $1$
Conductor $161$
Sign $0.440 - 0.897i$
Analytic cond. $17.3018$
Root an. cond. $17.3018$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.928 − 0.371i)2-s + (0.995 + 0.0950i)3-s + (0.723 − 0.690i)4-s + (0.327 − 0.945i)5-s + (0.959 − 0.281i)6-s + (0.415 − 0.909i)8-s + (0.981 + 0.189i)9-s + (−0.0475 − 0.998i)10-s + (0.928 + 0.371i)11-s + (0.786 − 0.618i)12-s + (−0.841 + 0.540i)13-s + (0.415 − 0.909i)15-s + (0.0475 − 0.998i)16-s + (−0.235 + 0.971i)17-s + (0.981 − 0.189i)18-s + (−0.235 − 0.971i)19-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)2-s + (0.995 + 0.0950i)3-s + (0.723 − 0.690i)4-s + (0.327 − 0.945i)5-s + (0.959 − 0.281i)6-s + (0.415 − 0.909i)8-s + (0.981 + 0.189i)9-s + (−0.0475 − 0.998i)10-s + (0.928 + 0.371i)11-s + (0.786 − 0.618i)12-s + (−0.841 + 0.540i)13-s + (0.415 − 0.909i)15-s + (0.0475 − 0.998i)16-s + (−0.235 + 0.971i)17-s + (0.981 − 0.189i)18-s + (−0.235 − 0.971i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.440 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.156056741 - 2.589950282i\)
\(L(\frac12)\) \(\approx\) \(4.156056741 - 2.589950282i\)
\(L(1)\) \(\approx\) \(2.520160422 - 0.9725819726i\)
\(L(1)\) \(\approx\) \(2.520160422 - 0.9725819726i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.928 - 0.371i)T \)
3 \( 1 + (0.995 + 0.0950i)T \)
5 \( 1 + (0.327 - 0.945i)T \)
11 \( 1 + (0.928 + 0.371i)T \)
13 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (-0.235 + 0.971i)T \)
19 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (-0.580 - 0.814i)T \)
37 \( 1 + (0.981 + 0.189i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.888 + 0.458i)T \)
59 \( 1 + (-0.0475 - 0.998i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (-0.723 + 0.690i)T \)
79 \( 1 + (-0.888 - 0.458i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (-0.580 + 0.814i)T \)
97 \( 1 + (0.654 + 0.755i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.17285591674477528693090787493, −26.65107557972610445720241474764, −25.32867603205780562682268231115, −25.080348804021384795041962566777, −23.98228201896467628694873999515, −22.62719813241753665859129257887, −22.04385966876620299446244484564, −21.003060866993008711005004323995, −20.028489826367198737780840085399, −19.04309612804813945585476815677, −17.82400117429048769847873738126, −16.5504964016649493192704368009, −15.30652070205579918722755586435, −14.50164845509284017052778138569, −13.99727436577887815750121571074, −12.87489094485002418861691462846, −11.71741686358654180693897179433, −10.38481662864961408981363867267, −9.095161101295159629682340379, −7.64705112127102041281615668987, −6.938938438312092787670521306920, −5.68991721721212571960082784118, −4.05776947197782622850419111102, −3.08240329202989892183507098370, −2.04358437821342322316046683444, 1.419298516419956846712287494088, 2.4269378513393974532969801401, 4.02740171308669419998963974599, 4.694701011967161733709639186989, 6.25591691318562584753793623890, 7.55841733416937112678641684756, 9.114575497546624071415021335611, 9.735545692432612197303578218613, 11.28644271736477364558106280904, 12.61521340941293415440603344359, 13.13515002084994002444060422916, 14.351435053734013575696985413773, 14.97216809707180658244764284500, 16.18717615928486486891608629572, 17.28247637259695095017837225436, 19.06683122480636499929618720516, 19.85785446907671612546488506153, 20.44792286090678003175462344895, 21.5885553317112559954439723240, 22.07427543128486182694793852918, 23.78007424412893886765479296399, 24.38603029829704159697818657177, 25.18009680785718395411292003897, 26.15969524252864385712888265707, 27.618886941407147798799364568122

Graph of the $Z$-function along the critical line